Computational Materials Science 115 (2016) 72–76

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Energy absorption induced oscillation of a rotating curved carbonnanotube in a nano bearing

http://dx.doi.org/10.1016/j.commatsci.2015.12.0280927-0256/� 2015 Elsevier B.V. All rights reserved.

⇑ Corresponding author at: Research School of Engineering, The AustralianNational University, Canberra, ACT 2601, Australia

E-mail address: kuncai99@163.com (K. Cai).

Zhaoliang Gao a, Haifang Cai b, Jing Wan b, Kun Cai b,c,⇑a Institute of Soil and Water Conservation, Northwest A&F University, Yangling 712100, ChinabCollege of Water Resources and Architectural Engineering, Northwest A&F University, Yangling 712100, ChinacResearch School of Engineering, The Australian National University, Canberra, ACT 2601, Australia

a r t i c l e i n f o

Article history:Received 15 November 2015Received in revised form 12 December 2015Accepted 17 December 2015Available online 2 February 2016

Keywords:Nano deviceCarbon nanotubeCurved tubeTransmission systemResonance

a b s t r a c t

In a nano bearing, a curved inner carbon nano tube (CNT) constrained by two short outer CNTs will havean oscillation along the curved axis of the tube when a specified rotational velocity is input on one end ofthe inner tube. It is found that the free end has periodic axial translational oscillation and the amplitudeof oscillation is very high when the frequency of the input rotational velocity is close to an eigen/reso-nance frequency of the system, i.e., energy absorption of the inner tube from the interaction betweenthe inner and outer tubes. Higher curvature of the inner tube leads to higher value of fundamentalfrequency of the system. The free end of the inner tube also has obvious torque oscillation. Both of theaxial translational oscillation and torque oscillation of the free end can be used as output signals ofthe system as working in a nano signal generator. The mid part of the inner tube, i.e., the part betweentwo outer tubes, has obvious in-plane vibration, which indicates that the present nano bearing is atwo-dimensional device.

� 2015 Elsevier B.V. All rights reserved.

1. Introduction Many experimentalists try to fabric such nano device [2,10–12].

With the decreasing of the sizes of components in a micro-electro-mechanical system (MEMS), both of theoretical study anddevelopment of experiment techniques on design of a nano-electro-mechanical system (NEMS) become urgent in recent years.Due to excellent physical properties, carbon nano structures, e.g.,carbon nanotube (CNT) [1–3], graphene sheet [4,5], carbon nano-scroll [6,7], etc., attract much attention in the studies. In particular,CNT-based oscillator/bearing/motor is one of the new conceptualdesigns of NEMS.

By experiments on axial tension of inner tube in multi-walledcarbon nanotubes (MWCNTs), Cumings and Zettl [2] test theinter-shell friction force and find that the extruded core can beretracted inside the outer tubes quickly when the core is released.They explain that the van der Waals force drives the retraction, anddemonstrate a possible path to construct a gigahertz-frequencyoscillator. Zheng and Jiang [8] propose the mechanical model ofsuch nano oscillator. Following the finding, a series of investiga-tions have been made to give a design of the nano device. Forexample, Legoas et al. [9] research the stability of the nano oscilla-tor with molecular dynamics (MD) simulations.

However, up to now, there is no one nano device with size of onlyfew nanometers. The reason is that the energy dissipation is seri-ous during relative motion between adjacent tubes and a long-time stable motion cannot be easily maintained. Using MD simula-tions, Zhao et al. [13] imply that the friction phenomenon betweentwo adjacent CNTs is the major reason for the transmission oforderly translational kinetic energy into disorderly thermal vibra-tional energy. As can be seen, the damping behavior still existseven in short-tube oscillators (see Fig. 1c in Ref. [13]). Guo et al.[14] compare the energy dissipation rates between commensurateand incommensurate double-walled CNT (DWCNT)-based oscilla-tors. Their results show that the inter-tube friction is proportionalto energy dissipation rate, but inversely proportional to the oscilla-tion frequency. In a long-tube oscillator, the damping oscillation isvery clear [15]. Cook et al. [16] find that friction within a rotatingCNT bearing is proportional to relative rotating speed and systemtemperature, but slightly depends on the length and meandiameter of inner tube. Hence, to obtain a nano oscillator/motorwith controllable amplitude or frequency, the energy dissipationdue to inter-tube friction should be made up by absorbing energyfrom environment. For example, Neild et al. [17] suggest to usingperiodic forces to excite the oscillation meeting the conditions ofcontrollability. Kang et al. [18] try to find a frequency-controlledCNT-based oscillator, which is made from three co-axial CNTs,

Z. Gao et al. / Computational Materials Science 115 (2016) 72–76 73

i.e., two outer tubes and one inner tube. By adjusting the gapbetween two outer tubes, the oscillation of the inner tube behavesdifferently. Either external electricity field or non-uniform heatingmethod can also be adopted to drive a nano rotary/linear motor.For example, Fennimore et al. [10] choose the MWCNTs as a keymotion-enabling component in a nano electricity-driven actuator.Bourlon et al. [11] attach a plate on the MWCNTs-based bearing,which is driven to rotate by a charged stator. By experiments, Bar-reiro et al. [12] find the axial motion of a cargo on a movable CNTrelative to another coaxial CNT with axial thermal gradient. Furtherstudy on thermal-gradient driven motion is given by otherresearchers [19–22]. Peculiarly, Cai et al. [23] find that the uniformtemperature field can also drive the rotation of a DWCNTs-basedmotor.

From the forgoing review, few work demonstrates the oscilla-tion and simultaneous rotation of the movable component in anano motor. Recently, Cai et al. [24] find that the high-speed rotat-ing inner tube in a fixed outer tube can be excited to oscillate peri-odically. The mechanism is that the rotational kinetic energy ofinner tube becomes into both of thermal vibration energy ofsystem and axial translational kinetic energy of inner tube. Thisfinding indicates that a stable oscillation can also be obtainedwhen the inner tube has a stable rotational speed.

Different from both of the work by Kang et al. [18] and Cai et al.[23], in the present study, we propose a new rotation–oscillationtransmission system from DWCNTs. But in the presentmodel, the inner tube is curved [25–32] and constrained in twoshort outer tubes. Giving a specified rotational speed on one endof the inner tube, the other tube end has different dynamicresponse. MD simulations are carried out to show the dynamicresponse.

Fig. 1. (a) The initial simulation model for a nano rotation–oscillation transmission systefixed. Initially, there are 16 layers of atoms on each end of the inner tube beyond the conscarbon nanotube has a constant input rotational speed, i.e., xin, and the upper right endbetween the upper right end of the inner tube and stator 2. Between the two outer tubesaxis, i.e., R, is the distance between O and O⁄. h is the central angle, and equal 90� in this swith the outer tubes and the included angle is �2� (see the sections within dashed boxreferred to the web version of this article.)

2. Models and methods

Three models for the system shown in Fig. 1b–d are considered:Model 1: curvature radius of R = 5.161 nm when n = 128; Model 2:R = 8.019 nmwhen n = 164; andModel 3: R = 11.194 nmwhen n = 204.

The molecular dynamic (MD) simulation is carried out usingLAMMPS [33]. In simulation, the AIREBO potential [34] is adoptedto indicate the interaction of the carbon atoms in system. Thetime integral increment is 1 fs. Before simulation, 400 ps ofNosé–Hoover bath at canonical (NVT) ensemble with T = 300 K isapplied on the system for relaxation. During relaxation, the top 4layers of the upper right end of the inner tube are fixed. The twoouter tubes are fixed, too. After relaxation, the input rotationalfrequency is applied on the lower left end of the curved innertube, and the value of gap, output rotational frequency and thepotential of inner tube are output.

To find the influence of the input rotational frequency of theinner tube on the dynamic response of system, the value of xin,is set to be 20, 50, 80, 120 and 150 GHz, respectively.

3. Results and discussions

From either potential history curve or output rotation historycurve in Table 1, one can find that the system tends to be stableafter �2 ns. The wild fluctuation of the output signals of systemin the initial stage is mainly due to the sudden application of theinput rotation of the curved inner tube. After 2 ns, most of the out-put signals keep stable. Two characters of the output signals arenecessary to be demonstrated. One is the damping behavior. Theother is the stable large amplitude vibration of the upper rightend of the curved inner tube.

m, in which the two straight 16-layer outer (10, 10) carbon tubes (orange parts) aretraint of the stators. The lower left end (grey part, 16-layer) of the curved inner (5, 5)has an output rotational frequency, i.e., xout. The value of gap is the axial distance, the mid part of the inner tube with n layers is curved. And the radius of the curvedtudy. After relaxation (b, c and d), near stator 1, the inner tubes do not keep co-axiales). (For interpretation of the references to color in this figure legend, the reader is

Table 1Dynamic response of the system with different model parameters and input rotational speeds (xin) within [0, 4] ns.

Gap history Potential energy history Output rotation history

0 1 2 3 41.82.12.41.82.12.41.82.12.41.82.12.41.82.12.4

Time (ns)

Gap

(nm

)

150GHz

120GHz

80GHz

50GHz

R = 5.161 nm 20GHz

0 1 2 3 4-9280-9270-9260-9270

-9260

-9265-9260-9255-9265-9260-9255-9265-9260-9255

R = 5.161 nm

Time (ns)Po

tent

ial e

nerg

y of

inne

r tub

e (e

V)

150GHz

120GHz

80GHz

50GHz

20GHz

0 1 2 3 40

150300

0120240

080

160

050

100

-800

80

150GHz

Time (ns)

120GHz

Out

put r

otat

iona

l fre

quen

cy (G

Hz)

80GHz

50GHz

20GHzR = 5.161 nm

0 1 2 3 4123123123123123

Time (ns)

Gap

(nm

)

150GHz

120GHz

80GHz

50GHz

20GHzR = 8.019 nm

0 1 2 3 4-11910-11900-11890-11910-11900-11890

-11900-11890-11880-11900-11890-11880-11895-11890-11885

150GHz

Time (ns)

120GHz

80GHz

Pote

ntia

l ene

rgy

of in

ner t

ube

(eV)

50GHz

20GHzR = 8.019 nm

0 1 2 3 40

150300

0120240

080

160

050

100

-800

80160

150GHz

Time (ns)

120GHz

80GHz

Out

put r

otat

iona

l fre

quen

cy (G

Hz) 20GHz

50GHz

R = 8.019 nm

0 1 2 3 41.62.43.21.62.43.21.62.43.21.62.43.21.62.43.2

150GHz

Time (ns)

120GHz

80GHz

Gap

(nm

)

50GHz

20GHzR = 11.194 nm

0 1 2 3 4-14840-14820-14800-14840-14820-14800

-14820-14810

-14820-14810

-14820-14810

Time (ns)

Pote

ntia

l ene

rgy

of in

ner t

ube

(eV)

120GHz

150GHz

80GHz

50GHz

R = 11.194 nm20GHz

0 1 2 3 40

150300

0120240

080

160-100

0100-80

080

150GHz

Time (ns)

120GHz

Out

put r

otat

iona

l fre

quen

cy (G

Hz)

80GHz

50GHz

20GHzR = 11.194 nm

74 Z. Gao et al. / Computational Materials Science 115 (2016) 72–76

As we know, the potential energy of the inner tube depends ontwo factors, i.e., the number of covalently bonded atoms and thedeformation of tube. Therefore, from the mid column in Table 1one can find that the potential fluctuates obviously in the first2 ns. The fluctuation is caused by two factors: the thermal vibra-tion of atoms, and the large deformation of the inner tube.

In the right column in Table 1, the value of output rotational fre-quency of the inner tube is also fluctuating during simulation. Butthe mean value (on time average) of the frequency always equalsthe input rotational frequency. It indicates that the inner tube isnot damaged under distortion. After 2 ns of simulation, the fluctu-ation of the output rotational frequency tends to be stable, and theamplitudes are�51.66, 26.22, 36.78, 29.63 and 31.50 GHz in Model1 with five different input rotational speeds (20, 50, 80, 120 and150 GHz), respectively. It implies that the upper right end of theinner tube has drastic torsional oscillation during simulation. ForModel 2 and Model 3, one can find the torsional oscillation, as well.

Before demonstrating the stable vibration of the gap (or theupper right end of the inner tube with large amplitude), the mech-anism for the vibration should be given. When the lower left end of

the curved inner tube rotates, the rest part of the curved tube is dri-ven to rotate because the strong interactionwithin the shell, i.e., thesp2 carbon–carbon bonds can keep the whole curved tube undam-aged. If the shell is curved, the axial/longitudinal distance betweenthe atoms on the shell varies periodically. Therefore, the vibration ofthe shell exists along its generatrix direction. If the two ends of thecurved inner tube are constrained rigidly, the amplitude of suchvibration will be negligible. However, in a real system, the curvedinner tube is constrained by two straight outer tubes. The interac-tion between the inner tube and stators is the van der Waals(vdW) force among the atoms on tubes. And the vdW force is sensi-tive to the relative positions of atoms. Simultaneously, the curvedtube tends to be straight (Fig. 1b, c and d). Hence, the curved innertube is subjected to a non-equilibrium transversal vdW force fromthe lower left outer tube. It leads to variation of the curvature ofthe inner tube locally. This is the reason for the vibration of theupper right end of the inner tube when a constant rotational speedis applied on the lower left end of the inner tube.

To demonstrate the vibration of the inner tube exited by inputrotation, Fig. 2 gives five snapshots of Model 2 (R = 8.019 nm) with

Fig. 2. Snapshots of Model 2 (R = 8.019 nm) andxin ¼ 50 GHz during [2.89, 2.91] ns. The top layer lists front views (in xz-plane) of configurations at different time. The lowerlayer lists vertical views (in yz-plane) of configurations.

Z. Gao et al. / Computational Materials Science 115 (2016) 72–76 75

xin ¼ 50 GHz. From 2.89 ns to 2.91 ns, the inner tube has a periodof vibration. Because the lower left end of the inner tube cannotmove axially and the axial length of the tube keeps approximatelyunchanged, the mid-part of the inner tube between the two fixedouter tubes has different length and different curvature duringvibration. Obviously, the included angle between this part andthe outer tube varies at the same time. Therefore, the vdW interac-tion between the inner tube and the outer tubes also shows peri-odic. The lower layer in Fig. 2 shows that the inner tube has veryslight vibration in yz-plane. It indicates that the inner tube behavesperfect toughness during rotation along its curved axis. The reasonis that the bending stiffness of the inner tube in yz-plane is muchgreater than the shearing stiffness of the tube in xz-plane in thepresent system. It also hints that the present nano bearing is atwo-dimensional device.

Meanwhile, the friction between the inner tube and the outertubes leads to the damping vibration of the curved inner tube(e.g., the history curve of gap at the situation of R = 8.019 nm andxin ¼ 50 GHz in Table 1).

When the system is in a stable state, the periodic vibration of gapcan be considered as an output signal of the system. And the ampli-tude of gap fluctuation, which can be considered as axial oscillationof the upper right end of the curved inner tube, in the same modelvaries with the value of xin. For example, in Fig. 3, which gives thedynamic response of the systemwithin [2.8, 3.0] ns, the amplitudesof gap inModel 1 (R = 5.161 nm) are�0.084, 0.044, 0.122, 0.063 and0.045 nm when xin ¼ 20; 50; 80; 120 and 150 GHz, respectively.Obviously, the amplitude with respect to xin ¼ 80 GHz in Model1 is the highest one among the five cases. Similarly, the amplitudesof gap inModel 2 (R = 8.019 nm) are�0.067, 0.710, 0.102, 0.013 and0.052 nm for five input rotational speeds, respectively. And

2.8 2.9 3.01.82.12.41.82.12.41.82.12.41.82.12.41.82.12.4

Time (ns)

Gap

(nm

)

150GHz

120GHz

80GHz

50GHz

(a) R = 5.161 nm 20GHz

2.8 2123123123123123

Tim

Gap

(nm

)

(b) R =

Fig. 3. History curves of gap and potential energy of the inner tube in the three models dR = 11.194 nm.

0.710 nm with respect to xin ¼ 50 GHz in Model 2 is the greatestamong the five cases. In Model 3 (R = 11.194 nm), the amplitudesof gap are �0.397, 0.203, 0.224, 0.605 and 0.243 nm for five inputrotational speeds, respectively. The amplitudes with respect toxin ¼ 20 and 120 GHz aremuch larger than those of the other threecases. It is concluded that the amplitude of gap fluctuation dependson the input rotational frequency and the value of R. Therefore, theinput rotational frequency of the curved inner tube which has thehighest amplitude of gap must be very close to one of the eigen/resonance frequencies of the dynamic system. And we callxin ¼ 80 GHz is an approximate eigen frequency (AEF) of Model 1(R = 5.161 nm), or xin ¼ 50 GHz of Model 2 (R = 8.019 nm). It isknown that the resonance of the inner tube is actually due to theenergy absorption of the inner tube from interaction between thetubes. Hence, oscillation of the upper right end of the inner tubeis caused by energy absorption. For Model 3 (R = 11.194 nm), bothof 20 and 120 GHz are as AEFs. One can find that the system withlower value of R has a higher value of AEF. The reason is that theinner tubes in the three models have the same chirality and theouter tubes are also identical, the stiffness of the systemwith lowervalue of R is higher and, therefore, the AEF is higher.

The curve (Model 2 with xin ¼ 50 GHz in Fig. 3) also indicatesthat the signal in the modulated carrier transmission (MCT) withperiod of �0.03 ns. The period is of �0.05 ns for the MCT (in therectangle frame) in Model 1 or Model 2 with xin ¼ 20 GHz.

Fig. 4a gives the histories of gap of Model 2 with different valuesof input rotational frequency. When the input rotational frequencyvaries from 42 to 58 GHz, the curves fluctuate differently. Theamplitude of gap with respect to 44 GHz is much lower than thatof 46 GHz curve. From 46 to 58 GHz, the amplitude decreasesmonotonously. To demonstrate the character of amplitudes of

.9 3.0

e (ns)

150GHz

120GHz

80GHz

50GHz

20GHz

8.019 nm

2.8 2.9 3.01.62.43.21.62.43.21.62.43.21.62.43.21.62.43.2

150GHz

Time (ns)

120GHz

80GHz

Gap

(nm

)

50GHz

20GHz

(c) R = 11.194 nm

uring [2.8, 3.0] ns. (a) Model 1, R = 5.161 nm; (b) Model 2, R = 8.019 nm; (c) Model 3,

(b) Amplitude(a) History during [2.8, 3.0] ns

2.8 2.9 3.0020202020202020202

Time (ns)

42GHz

44GHz

46GHz

48GHz

Gap

(nm

)

50GHz

52GHz

54GHz

56GHz

58GHz

40 45 50 55 60 65

0.3

0.6

0.9

1.2

(42,0.181)

(44,0.381)

(46,1.122)

(48,0.892)

(50,0.71)

(52,0.477)(54,0.441)

(56,0.228)(58,0.176)

Ampl

itude

of g

ap (n

m)

Input rotational frequency (GHz)

Fig. 4. Histories of gap and amplitudes of gap in Model 2 with respect to input rotational frequency during [2.8, 3.0] ns.

76 Z. Gao et al. / Computational Materials Science 115 (2016) 72–76

gap in Model 2 with different input rotational frequencies, wecalculate the average amplitudes of gap during [2.8, 3.0] ns(in Fig. 4a) and show them in Fig. 4b. As can be seen, the maximumof amplitude is 1.122 nm with respect to xin ¼ 46 GHz, which isobviously larger than 0.710 nm with respect to 50 GHz. Hence,we conclude that 46 GHz is much closer to the eigen frequencythan 50 GHz.

4. Conclusions

From numerical computation, the oscillation behavior of thecurved rotary inner tube in the bearing from CNTs is investigated.Some remarkable conclusions can be drawn.

(1) Inputting a rotational frequency on one end of the curvedinner tube, the other end (free end) of the inner tube willoutput not only the rotation but also the axial translationaloscillation and torque oscillation.

(2) The amplitude of axial translational oscillation depends onthe difference between the input rotational frequency andthe eigen/resonance (serious energy absorption state) fre-quency of the system, lower difference leads to higheramplitude. Higher curvature of the mid part of the innertube has higher fundamental frequency.

(3) The oscillation of the inner tube happens within the planewhere the curved inner tube is layout. Hence, the nano bear-ing is a two-dimensional device.

(4) The output signals, e.g., oscillation and rotation of the freeend of the inner tube, behave as MCTs, which implies apotential application in design of a nano signal generator.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.commatsci.2015.12.028.

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